DEPARTMENT  OF  THE  INTERIOR 

BUREAU  OF  EDUCATION 


BULLETIN,  1920,  No.  1 


THE  PROBLEM  OF  MATHEMATICS  IN 
SECONDARY  EDUCATION 


A  REPORT  OF  THE  COMMISSION  ON 
THE  REORGANIZATION  OF  SECOND- 
ARY EDUCATION,  APPOINTED  BY  THE 
NATIONAL  EDUCATION  ASSOCIATION 


WASHINGTON 

GOVERNMENT  PRINTING  OFFICE 
1920 


DEPARTMENT  OF  THE  INTERIOR 

BUREAU  OF  EDUCATION 


BULLETIN,  1920,  No.  I 


THE  PROBLEM  OF  MATHEMATICS  IN 
SECONDARY  EDUCATION 


A  REPORT  OF  THE  COMMISSION  ON 
THE  REORGANIZATION  OF  SECOND- 
ARY EDUCATION,  APPOINTED  BY  THE 
NATIONAL  EDUCATION  ASSOCIATION 


WASHINGTON 

GOVERNMENT  PRINTING  OFFICE 
1920 


ADDITIONAL  COPIES 

OP   THIS  PUBLICATION  MAY  BE  PROCURED   FROM 

THE  SUPERINTENDENT  OF  DOCUMENTS  , 

GOVERNMENT  PRINTING  OFFICE 

WASHINGTON,  D.  C. 

AT 

5  CENTS  PER  COPY 


REPORTS  OF  THE  COMMISSION  ON  THE  REORGANIZATION  OF  SEC- 
ONDARY EDUCATION. 

The  following  reports  of  the  commission  have  been  issued  as  bulletins  of  the 
United  States  Bureau  of  Education  and  may  be  procured  from  the  Superin- 
tendent of  Documents,  Government  Printing  Office,  Washington,  D.  C.,  at  the 
prices  stated.  Remittance  should  be  made  in  coin  or  money  order.  Other 
reports  of  the  commission  are  in  preparation. 

1915,  No.  23.  The  Teaching  of  Community  Civics.     10  cents. 

1916,  No.  28.  The  Social  Studies  in  Secondary  Education.     10  cents. 

1917,  No.   2.  Reorganization  of  English  in  Secondary  Schools.    20  cents. 
1917,  No.  49.  Music  in  Secondary  Schools.     5  cents. 

1917,  No.  50.  Physical  Education  in  Secondary  Schools.     5  cents. 

1917,  No.  51.  Moral  Values  in  Secondary  Education.     5  cents. 

1918,  No.  19.  Vocational  Guidance  in  Secondary  Schools.     5  cents. 

1918,  No.  35.  Cardinal  Principles  of  Secondary  Education.     5  cents. 

1919,  No.  55.  Business  Education  in  Secondary  Schools.     10  cents. 

1920,  No.    1.  The  Problem  of  Mathematics  in  Secondary  Education.    5  cents. 


COMMITTEE  ON  THE  PROBLEM  OF  MATHEMATICS  IN  SECONDARY 

EDUCATION. 

William   H.   Kilpatrick,   chairman,  professor  of  education,   Teachers  College, 

Columbia  University,  New  York,  N.  Y. 
Fred  R.  Hunter,  superintendent  of  schools,  Oakland,  Calif. 
Franklin  W.  Johnson,  principal  University  High  School,  University  of  Chicago, 

Chicago,  111. 
J.    H.    Miunick,    assistant   professor   of   educational    methods,    University   of 

Pennsylvania,  Philadelphia,  Pa. 

Raleigh  Shorling,  Lincoln  School,  646  Park  Avenue,  New  York,  N.  Y. 
J.  C.  Stone,  head  of  department  of  mathematics,  State  Normal  School,  Mont- 

clair,  N.  J. 

Milo  H.  Stuart,  principal  Technical  High  School,  Indianapolis,  Ind. 
J.  H.  Withers,  superintendent  of  schools,  St.  Louis,  Mo. 


THE    REVIEWING    COMMITTEE    OF    THE    COMMISSION    ON    THE 
REORGANIZATION  OF   SECONDARY  EDUCATION. 

(The  Reviewing  Committee  consists  of  26  members,  of  whom  16  are  chairmen  of  com- 
mittees and  10  are  members  at  large.) 

Chairman  of  the  Commission  and  of  the  Rcvieicring  Committee: 

Clarence  D.  Kingsley,  State  high-school  supervisor,  Boston,  Mass. 
Members  at  large: 

Hon.  P.  P.  Claxton,  United  States  Commissioner  of  Education,  Washing- 
ton, D.  C. 

Thomas  H.  Briggs,  associate  professor  of  education,  Teachers  College,  Co- 
lumbia University,  New  York,  N.  Y. 

3 


MATHEMATICS  IN  SECONDARY  EDUCATION. 


Members  at  Ivrn? 

Alexander  Inglis,  assistant  professor  of  education,  in  charge  of  secondary 

education,  Harvard  University,  Cambridge,  Mass. 
Henry  Neumann,  Ethical  Culture  School,  New  York,  N.  Y. 
William  Orr,  senior  educational  secretary,  international  Y.  M.  C.  A.  com- 

mittee, 347  Madison  Avenue,  New  York,  N.  Y. 
Willialu  B.  Owen,  principal  Chicago  Normal  College,  Chicago,  111. 
Edward  O.  Sisson,  president  University  of  Montana,  Missoula,  Mont. 
Joseph  S.  Stewart,  professor  of  secondary  education,  University  of  Georgia, 

Athens,  Ga. 

Milo  H.  Stuart,  principal  Technical  High  School,  Indianapolis,  Ind. 
H.  L.  Terry,  State  high-school  supervisor,  Madison,  Wis. 
Chairmen  of  Committees: 

Administration  of  Secondary  Education  —  Charles  H.  Johnston,  professor  of 

secondary  education,  University  of  Illinois,  Urbana,  111.1 
Agriculture  —  A.  V.  Storm,  professor  of  agricultural  education,  University  of 

Minnesota,  St.  Paul,  Minn. 
Art  Education—  Royal  B.  Farnuin,  president,  Mechanics  Institute,  Rochester, 

N.  Y. 
Articulation  of  High  School  and  College  —  Clarence  D.  Kingsley,  State  high- 

school  supervisor,  Boston,  Mass. 
Business  Education  —  Cheesman  A.  Herrick,  president,  Girard  College,  Phil- 

adelphia, Pa. 
Classical  Languages  —  Walter  Eugene  Foster,  Stuyvesant  High  School,  New 

York,  N.  Y. 

English—  James  Fleming  Hosic,  Chicago  Normal  College,  Chicago,  111. 
Household  Arts  —  Mrs.  Henrietta  Calvin,  United  States  Bureau  of  Educa- 

tion, Washington,  D.  C. 
Industrial  Arts  —  Wilson  H.  Henderson,  extension  division,  University  of 

Wisconsin,  Milwaukee,  Wis.    (now  Major,  Sanitary  Corps,  War  Depart- 

ment, U.  S.  A.). 
Mathematics—  William  Heard  Kilpatrick,  associate  professor  of  education, 

Teachers  College,  Columbia  University,  New  York,  N.  Y. 
Modern  Languages  —  Edward  Manley,  Englewood  High  School,  Chicago,  111. 
Music  —  Will  Earhart,  director  of  music,  Pittsburgh,  Pa. 
Physical  Education  —  James  H.  McCurdy,  director  of  normal  courses  of 
'    physical  education,  International  Y.  M.  C.  A.  College,  Springfield,  Mass. 
Sciences  —  Otis  W.  Caldwell,  director,  Lincoln  School,  and  professor  of  edu- 

cation, Teachers  College,  Columbia  University,  New  York,  N.  Y. 
Social  Studies  —  Thomas  Jesse  Jones,  United  States  Bureau  of  Education, 

Washington,  D.  C. 
Vocational    Guidance  —  Frank    M.    Leavitt,    associate    superintendent    of 

schools,  Pittsburgh,  Pa. 

»  Deceased  Sept  4,  1917. 


CONTENTS. 


Page. 

Membership  of  the  committee  on  the  problem  of  mathematics 3 

Membership  of  the  reviewing  committee  of  the  commission 3 

Letter  of  transmittal 7 

Preface 8 

I.  Introduction 9 

II.  The  demand  for  an  inquiry . ^ 9 

III.  Analysis  of  the  situation 11 

1.  The  problem  of  presentation 11 

2.  The  several  needs  for  mathematics 14 

3.  Comparative  values 15 

4.  Formal   discipline 16 

5.  The  needs  of  the  several  groups 17 

6.  Selecting  mathematical  ability 20 

IV.  Suggestions  as  to  courses . 21 

1.  The  work  of  the  junior  high  school 22 

2.  Trade  mathematics 23 

3.  Preliminary  engineering 23 

4.  For  the  specializes 23 

5 


LETTER  OF  TRANSMITTAL. 


DEPARTMENT  OF  THE  INTERIOR, 

BUREAU  OF  EDUCATION, 
Washington,  October  11,  1919. 

SIR:  One  of  the  committees  of  the  Commission  on  the  Reorgani- 
zation of  Secondary  Education,  appointed  by  the  National  Educa- 
tion Association,  and  several  of  whose  reports  this  bureau  has 
already  published  in  the  form  of  bulletins,  undertook  the  study  of 
mathematics  in  the  high  schools.  As  stated  by  this  committee  in  the 
introduction  to  this  report  and  by  the  chairman  of  the  commission 
in  the  preface,  the  committee  found  itself  unable  to  make  final  recom- 
mendations in  regard  to  the  reconstruction  of  the  courses  of  study  in 
this  subject  in  the  high  schools.  The  committee  has,  therefore,  con- 
fined its  work  to  a  preliminary  report,  presenting  an  analysis  of  the 
subject,  and  raising  certain  fundamental  questions  which  must  be 
answered  before  the  reconstruction  desired  can  be  undertaken  intelli- 
gently and  with  any  certainty  of  satisfactory  success. 

I  am  transmitting  this  preliminary  report  for  publication  as  a 
bulletin  of  the  Bureau  of  Education,  in  order  that  in  this  form  it 
may  be  accessible  to  students  of  education,  teachers  of  mathematics, 
and  directors  of  mathematics  teaching  in  high  schools.  It  is  ex- 
pected that  it  will  give  rise  to  such  discussion  and  experimenting 
as  will  enable  other  committees  to  carry  forward  the  work  of  the 
point  of  definite  reconstruction  of  courses  of  study  in  this  subject 
for  the  several  classes  of  high-school  pupils. 
Respectfully  submitted. 

P.  P.  CLAXTON, 

Commissioner. 
The  SECRETARY  OF  THE  INTERIOR. 

7 


PREFACE. 


The  Commission  on  the  Reorganization  of  Secondary  Education 
finds  itself  confronted  with  problems  of  great  difficulty  in  recom- 
mending a  reorganization  of  the  mathematical  studies  of  the  sec- 
ondary school.  Antecedent  to  new  courses,  there  should  be  an  agree- 
ment among  psychologists  and  educators  such  as  has  not  yet  been 
reached.  It  seems,  therefore,  that  the  best  service  that  the  commis- 
sion can  at  this  time  render  is  to  present  an  analysis  of  the  situation. 
This  report,  therefore,  is  submitted  primarily  for  the  purpose  of 
stimulating  discussion.  It  is  hoped  that  the  practical  suggestions 
will  also  serve  to  direct  experimentation  in  planning  new  courses  for 
secondary  school  students  of  the  various  types  here  recognized. 

CLARENCE  D.  KINGS:LEY, 

Chairman  of  the  commission. 
3 


THE  PROBLEM  OF  MATHEMATICS  IN  SECONDARY  EDUCATION. 


I.  INTRODUCTION. 

Few  subjects  taught  in  the  secondary  school  elicit  more  contra- 
dictory statements  of  view  than  does  mathematics.  What  should  be 
taught,  how  much  of  it,  to  whom,  how,  and  why,  are  matters  of  dis- 
agreement. There  is  every  variety  of  position.  A  conservative  group 
would  keep  substantially  unchanged  the  customary  content  and  divi- 
sion into  courses,  and  find  the  hope  of  improvement  in  a  more  ade- 
quate preparation  of  teachers.  To  this  limited  reform  an  increasing 
number  object,  with  little  agreement,  however,  among  themselves. 
Amid  the  conflict  of  opinions  the  committee  on  the  problem  of  mathe- 
matics in  secondary  education  believes  that  a  reconsideration  of  the 
whole  question  is  desirable. 

To  present  the  finished  details  of  a  working  plan  would  have 
been  most  gratifying  to  the  committee,  but  this  has  been  judged  im- 
possible. The  situation  seems  to  force  the  limitation.  To  carry 
weight,  such  a  detailed  plan  would  have  to  be  based  upon  a  wider 
range  of  experiment  than  in  fact  exists.  Only  recently  has  there 
been  serious  effort  to  consider  the  problem  of  the  proper  content 
and  arrangement  of  the  courses  in  secondary  mathematics.  The 
pertinent  experiments  available  for  study  do  not  as  yet  present  a 
variety  of  type  and  testing  sufficient  to  establish  the  necessary 
conclusions.  Within  the  time  allotment  available  to  the  committee 
there  seemed  then  only  the  choice  between  no  report  and  an  admit- 
tedly preliminary  report.  The  committee  has  chosen  the  latter  alter- 
native, and  proposes  to  lay  before  the  American  educational  public 
(1)  some  of  the  considerations  that  demand  a  fresh  study  of  the 
problems  involved,  (2)  some  of  the  factors  that  bear  upon  the  solu- 
tion of  the  problem,  and  (3)  certain  tentative  suggestions  for  ex- 
perimentation to  develop  new  and  better  courses.1  It  is  but  fair  to 
say  that  few  of  the  specific  suggestions  made  are  in  fact  new,  many 
being  already  somewhere  actually  in  practice. 

II.  THE  DEMAND  FOR  AN  INQUIRY. 

An  inquiry  into  the  advisability  of  reorganizing  and  reconstitut- 
ing secondary  mathematics  is  demanded  from  a  variety  of  considera- 
tions. 

1  It  is  gratifying  to  note  that  the   Mathematical   Association  of  America  is  pushing  a 
program  of  study  and  experimentation  along  lines  quite  similar  to  those  here  discussed. 
155900°— 20 2  9 


10  .  ,  .  t MATHEMATICS  IN   SECONDARY  EDUCATION. 

1.  It]  is  be in^  insisted  as  never  before  that  each  subject  and  each 
fern JS^t  tJje  subject  jivstify  itself;  or,  negatively,  that  no  subject  or 
item  be  retained  in  any  curriculum  unless  its  value,  viewed  in  rela- 

\  tion  to  other  topics  and  to  time  involved,  can  be  made  reasonably 
\  probable.  No  longer  should  the  force  of  tradition  shield  any  sub- 
ject from  this1,  scrutiny.  A  better  insight  into  the  conditions  of  social 
welfare,  and  the  many  changes  among  these  conditions,  alike  make 
inherently  probable  a  different  emphasis  upon  materials  in  the 
curriculum,  if  not  a  different  selection  of  actual  subject  matter. 
This  calls  for  a  review  and  revaluation,  in  particular,  of  all  our 
older  studies,  mathematics  not  least. 

f  2.  Moreover,  a  growing  science  of  education  has  come  to  place 
appreciably  different  values  upon  certain  psychological  factors  in- 
\volved,  chief  among  which  is  that  relating  to  "  mental  discipline." 
No  one  inclusive  formulation  of  the  older  position  can  be  asserted, 
yet  on  the  whole  there  was  acceptance  of  the  "  faculty  "  psychology 
with  an  uncritical  belief  in  the  possibility  of  a  good-for-all  training 
of  the  several  "  faculties."  To  the  extremist  of  this  school  the 
"  faculty  of  reasoning,"  for  example,  could  be  trained  on  any  ma- 
terial where  reasoning  was  involved  (the  more  evident  the  reasoning, 
the  better  the  training) ,  and  any  facility  of  reasoning  gained  in  that 

.  particular  activity,  could,  it  was  thought,  be  accordingly  directed  at 
will  with  little  loss  of  effectiveness  to  any  other  situation  where 
good  reasoning  was  desired.  In  probably  no  study  did  this  older 
doctrine  of  "mental  discipline"  find  larger  scope  than  in  mathe- 
matics, in  arithmetic  to  an  appreciable  extent,  more  in  algebra,  most 
of  all  in  geometry. 

AVith  the  scientific  scrutiny  of  the  conditions  under  which  "  trans- 
fer" of  training  takes  place,  the  inquiry  grows  continually  more 
insistent  as  to  whether  our  mathematical  courses  should  continue  un- 
changed, now  that  so  much  of  their  older  justification  has  been 
modified.  Possibly  both  purpose  and  content  need  to  be  changed. 

3.  Yet  another  demand  for  reconstruction  is  found  in  the  now 
generally  accepted  belief  that  not  all  high-school  pupils  should  take 
the  same  studies.     The  fact  of  marked  individual  differences  has 
been  scientifically  established.    The  principles  of  adaptation  to  such 
individual  differences,  that  is,  to  individual  needs  and  capacities,  is 
now  widely  accepted  in  the  high  schools  of  America.    The  exception 
calls  for  scrutiny.    Traditionally,  algebra  and  geometry  have  been 
required  for  graduation.     Is  this  necessary  or  advisable?     In  this 
growing  practice  of  differentiation  and  adaptation  we  have  then  a 
third  reason  for  at  least  reconsidering  the  customary  mathematics 
courses. 

4.  A  demand  for  reconsideration  well  worthy  of  our  attention  is 
found  in  the  insistent  question  whether  a  content  chosen  to  furnish 


ANALYSIS  OF  THE  SITUATION".  11 

preparation  for  further  but  remote  study  does  necessarily  or  even 
probably  include  the  wisest  selection  of  knowledge  useful  for  those 
who  do  not  reach  that  advanced  stage  of  study.  Whether  all  should 
learn  first  the  more  assuredly  useful  topics,  or  whether  alternative 
courses  should  be  offered,  are  proper  subjects  of  inquiry.  In  either 
event  we  find  in  this  consideration  a  fourth  reason  for  studying  anew 
the  offerings  of  our  high-school  mathematics. 

5.  A  fifth  reason  for  reconsideration  is  found  in  the  problem  of 
method.  Educators  are  studying  now  with  new  zeal  the  proper  pres- 
entation of  subject  matter  in  all  school  work.  Should  not  this  study 
extend  to  secondary  mathematics?  Have  we  arranged  the  subject 
matter  of  that  field  in  the  best  form  for  appropriation?  Might  it 
even  be  possible  that  mathematics  should  be  reorganized  in  a  way  to 
run  across  customary  lines  of  division?  Or  might  this  be  true  of 
some  parts  of  mathematics  for  some  groups  of  pupils  and  not  be  true 
of  all  ?  The  proper  answers  to  such  questions  are  not  at  once  evident, 
but  certainly  there  is  enough  point  in  the  inquiry  to  add  a  fifth 
reason  for  our  proposed  investigation. 

III.  ANALYSIS  OF  THE  SITUATION. 

1.  The  problem  of  presentation. — Far-reaching  differences  of 
method  carry  with  them  widely  different  organizations  of  subject 
matter,  especially  in  introductory  courses.  From  this  consider ation, 
at  least,  there  are  certain  advantages  in  discussing  as  the  first  factor 
in  the  situation  the  problem  of  presentation. 

The  traditional  school  method  has  been  that  based  upon  the  "  logi- 
cal" arrangement  of  subject  matter.  Thus  our  fathers  studied 
English  grammar  before  they  took  up  composition,  the  "science" 
being  "  logically  "  anterior  to  the  "  art."  The  science,  in  this  case 
grammar,  began  with  a  definition  of  itself  and  the  analysis  of  the 
subject  into  its  four  principal  divisions.  Then  came  the  definitions 
of  the  "  parts  of  speech."  It  was  a  long — and  generally  dreary — road 
before  the  pupil  could  see  any  bearing  of  what  he  learned  upon  any- 
thing else.  At  length,  after  toilsome  memorizing,  there  appeared 
within  the  subject  itself  a  new  variety  of  mental  gymnastics  which 
called  forth  from  some  a  certain  show  of  activity.  In  the  end  the 
survivors  caught  some  glimpse  of  what  it  had  all  been  about.  But 
when  they  took  up  the  "  art "  of  composition,  the  "  science  "  proved 
of  small  assistance.  Somehow  the  "  art "  had  to  be  learned  as  if  it 
alone  faced  the  actual  demand. 

From  an  implicit  reliance  upon  this  "logical"  arrangement  there 
has  come  a  revolt,  not  yet  universal,  but  still  unmistakably  at  hand. 
The  demand  has  now  become  insistent  that  in  arranging  subject 
matter  for  learning,  consideration  be  given,  not  to  "logic"  as 
formerly  conceived,  but  to  economy  in  learning  and  effective  con- 


12  MATHEMATICS  IN  SECONDARY  EDUCATION. 

trol  of  subject  matter.  This  reversal  of  method,  coupled  with  a  dis- 
trust of  the  theory  of  discipline,  has  thus  not  only  reduced  grammar 
to  a  small  fraction  of  its  former  self,  but  has,  besides,  greatly  re- 
arranged and  rewritten  the  study. 

Keeping  before  us  the  demand  for  economy  in  learning  and  effec- 
tive control  of  subject  matter,  what  can  we  say  about  method?  How 
does  learning  in  fact  take  place?  (1)  Eepetition  is  a  factor  in 
learning  known  to  all.  (2)  An  inclusive  "  set "  which  shall  predis- 
pose the  attention,  focus  available  inner  resources,  and  secure  repe- 
tition is  a  necessary  condition  less  commonly  considered.  (3)  The 
effect  of  accompanying  satisfaction  to  foster  habit  formation  is  a 
third  factor  to  be  noted.1  These  three  factors  are  necessary,  then,  to 
adequate  consideration  of  the  problem  of  method.  It  accords  with 
these  considerations  and  with  undisputed  observation  that,  other 
things  being  equal,  any  item  is  more  readily  learned  if  its  bearing 
and  need  are  definitely  recognized.  The  felt  need  predisposes  at- 
tention, calls  into  play  accessory  mental  resources,  and  in  proportion 
to  its  strength  secures  the  necessary  repetition.  As  the  need  is  met, 
satisfaction  ensues.  All  factors  thus  cooperate  to  fix  in  place  the 
new  item  of  knowledge.  The  element  of  felt  need  thus  secures  not 
only  the  learning  of  the  new  item,  but  it  has  at  the  same  time  called 
into  play  the  allied  intellectual  resources  so  that  new  and  old  are 
welded  together  in  effective  organization  with  reference  to  the  need 
which  originally  motivated  the  process. 

Lest  some  should  fear  that  by  need  is  here  meant  a  mere  "  bread 
and  butter  demand,"  the  committee  hastens  to  say  that  it  is  psycho- 
logic and  not  economic  need  which  acts  as  the  factor  in  learning. 
Economic  need  may  indeed  be  felt;  and,  if  so,  may  then  serve  to  in- 
fluence learning;  but  there  is  nothing  in  the  foregoing  argument  to 
deny  that  a  purely  "theoretic"  interest  might  not  be  as  potent  as 
any  other  to  bring  about  the  learning  and  organization  of  subject 
matter. 

To  speak  of  the  bearing  and  netd  of  any  new  material  is  to  imply 
the  presence  and  functioning  of  already  existent  purposes  and  in- 
terests. From  this  consideration  thus  related  to  the  foregoing  the 
committee  believes  that,  speaking  generally,  introductory  mathe- 
matics— ordinarily  conceived  as  separate  courses  in  algebra,  geome- 
try, and  trigonometry — should  be  given  in  connection  with  the 
solving  of  problems  and  the  executing  of  projects  in  fields  where  the 
pupils  already  have  both  knowledge  and  interest.  This  would  make 
the  study  of  mathematics  more  nearly  approximate  a  laboratory 
course,  in  which  individual  differences  could  be  considered  and  the 
effective  devices  of  supervised  study  be  utilized.  The  minimum  of 

1Tlvi  behaviorist  psychologist  by  definition  rejects  the  subjective  connotation  of  "  satis- 
faction." If  we  had  access  to  the  actual  psychology  involved,  possibly  the  difference  of 
statement  would  in  effect  disappear. 


ANALYSIS  OF  THE  SITUATION.  13 

tlie  course  might  well  in  this  way  be  cared  for  in  the  recitation 
period,  reserving  the  outside  work  rather  for  allied  projects  and 
problems  in  which  individual  interests  and  capacities  were  promi- 
nent factors. 

The  significant  element  in  this  conception  is  the  utilization  of 
ideas  and  interests  already  present  with  the  pupils  as  a  milieu 
within  which  the  mathematical  conception  or  process  to  be  taught 
finds  a  natural  setting,  and  from  which  a  need  to  use  the  conception 
or  process  can  as  a  consequence  be  easily  developed.  Where  this 
state  of  affairs  exists,  the  bearing  and  felt  need  utilize  the  laws  of 
learning  as  was  discussed  above,  and  the  mathematical  knowledge  or 
skill  is  fixed  in  a  manner  distinctly  economical  as  regards  both 
present  effort  and  future  applicability. 

As  was  stated  at  the  outset,  this  suggested  procedure  reaches 
beyond  the  questions  of  economy  of  learning  and  application — con- 
trolling though  these  here  are — to  the  question  of  content.  The  pro- 
cedure here  contemplated  makes  definite  demand  for  an  appropriate 
introductory  content.  To  work  along  this  line  there  must  be  made  a 
selection  of  conceptions  and  processes  which  can  serve  the  pupils  as 
instruments  to  the  attainment  of  the  ends  set  before  them  in  the  proj- 
ects or  problems  upon  which  they  are  at  work.  This  instrumental 
character  becomes  then  the  essential  factor  in  any  introductory 
course.  It  is  these  instrumental  needs  and  not  "  logical "  intercon- 
necteclness  which  must  give  unity  to  such  a  course.  A  content  thus 
instru mentally  selected  will,  on  the  one  hand,  be  free  of  the  old 
formal  puzzles,  the  complex  instances,  the  verbal  problems  which  in 
the  past  have  wasted  so  much  time  and  destroyed  so  much  potential 
interest;  and  will,  on  the  other,  run  across  the  divisions  heretofore 
separating  algebra,  geometry,  and  trigonometry. 

A  distinct  advantage  in  the  procedure  here  suggested  is  the  better 
promise  it  holds  out  of  meeting  in  one  introductory  course  the  needs 
of  both  those  who  will  go  on  to  advanced  study  in  mathematical  lines 
and  those  who  will  not.  Where  the  basis  of  selection  and  procedure 
is  instrumental,  all  can  begin  together.  The  future  specializes  in 
mathematics  will  as  the  course  proceeds  take  increasing  interest  in 
the  mathematical  relationships  involved  and  will  stress  this  aspect  in 
their  individual  problems  and  projects.  Those  whose  tastes  and 
aptitudes  lead  them  elsewhere  will  in  the  meanwhile  have  had  the 
opportunity  to  learn  in  practical  situations  some  of  the  mathematical 
concepts  and  processes  which  they  will  later  use  in  their  own  chosen 
fields.  Their  individual  projects  in  the  course  can  serve  well  as  con- 
necting links  between  the  mathematics  taught  and  their  later  field 
of  vocational  application. 

After  the  introductory  course  has  been  completed,  and  differenti- 
ation has  begun,  the  same  principles  still  hold,  though  in  the  different 


14  MATHEMATICS  IN  SECONDAKY  EDUCATION". 

fields.  Those  who  have  chosen  to  continue  the  study  of  mathematics 
as  such  will  find  their  problems  or  projects  within  the  field  of 
mathematics  itself,  quite  likely  examining  anew  in  the  light  of  wider 
acquaintance  assumptions  freely  made  in  the  earlier  period.  Eu- 
clid's system  of  axioms  and  postulates  might  here  receive  its  first 
careful  consideration.  Those  who  had  elected  to  prepare  for  engi- 
neering and  the  like  might  continue  to  find  their  mathematics  in  con- 
nection with  problems  or  projects  -devoted  now  particularly  to  a 
preliminary  engineering  content.  Conceptions  usually  reserved  for 
college  analytics  and  calculus — if  not 'indeed  already  used  in  the 
introductory  course — can  well  have  a  place  here.  Their  rich  in- 
strumental character  will  justify  their  presence,  even  if  they  lack 
somewhat  in  relationship  to  a  fully  developed  logical  system. 

2.  The  several  needs  for  mathematics — Among  the  multiplicity  of 
specific  occasions  for  using  mathematics  and  among  the  various  types 
of  subject  matter,  there  are  certain  possible  groupings  which  promise 
aid  in  the  determination  of  the  mathematical  courses. 

Without  implying  the  possibility  always  of  sharp  differentiation, 
we  may  distinguish  in  the  realm  of  mathematical  knowledge  (i) 
those  items  the  immediate  use  of  which  involve  a  minimum  of  think- 
ing, as,  for  example,  adding  a  column  of  figures,  and  (ii)  those  items 
which  are  primarily  used  as  notions  or  concepts  in  the  furtherance  of 
thinking.  It  is  clear  that  the  distinction  here  is  of  the  way  in  which 
the  knowledge  is  used  and  not  of  the  knowledge  itself ;  for  any  item 
of  knowledge  might  at  one  time  serve  one  function  and  at  another 
time  the  other.  It  would  still  remain  true,  however,  that  certain 
groups  of  people  might  have  characteristically  different  needs  along 
the  two  lines.  Under  the  first  head  we  should  include  the  mechanic's 
use  of  a  formula,  the  surveyor's  use  of  his  tables,  the  statistician's 
finding  of  the  quartile.  The  man  in  the  street  would  call  this  the 
"practical''  usp  of  mathematics.  Under  the  other  head  we  should  in- 
clude the  intelligent  reader's  use  of  mathematical  language  by  which 
he  would  understand  an  account  of  Kepler's  three  famous  laws. 
Some  may  wisli  to  call  this  the  "cultural"  use  of  mathematics.  The 
term  "interpretative"  might,  however,  more  exactly  express  the  dif- 
ferentiating idea. 

We  may  next  ask  whether  there  are  different iable  groups  among' 
high-school  pupils  whose  probable  destinations  or  activities  deter- 
mine within  reasonable  limits  the  extent  and  type  of  their  future 
mathematical  needs.  In  a  democracy  like  ours,  -questions  of  prob- 
able destination  are  of  course  very  difficult.  There  must  be  110 
caste-like  perpetuation  of  economic  and  cultural  differences;  and 
definite  effort  must  be  made  to  keep  wide  open  the  door  of  further 
study  for  those  who  may  later  change  their  minds.  But  differ- 
entiating choices  are  in  fact  made;  and  in  view  of  the  wealth  of 


ANALYSIS   OF  THE   SITUATION.  15 

offerings  on  the  one  hand  and  of  individual  differences  on  the  other, 
such  choices  must  be  made.  Properly  safeguarded  by  an  intelligent 
effort  to  adopt  social  demands  to  individual  taste  and  aptitude,  these 
choices  should  work  to  the  advantage  both  of  the  individual  and  of 
the  group.  The  committee  considers  that  four  groups  of  users  of 
mathematics  may  be  distinguished : 

(a)  The  "  general  readers,"  who  will  find  their  use  of  mathematics 
beyond  arithmetic  confined  largely  to  the  interpretative  function 
described  above. 

(6)  Those  whose  work  in  certain  trades  will  make  limited,  but 
still  specific,  demand  for  the  "  practical "  use  of  mathematics. 

(c)  Those  whose  practical  work  as  engineers  or  as  students  of 
certain  sciences  requires  considerable  knowledge  of  mathematics. 

(d)  Those  who  specialize  in  the  study  of  mathematics  with  a 
view  either  to  research  or  to  teaching  or  to  the  mere  satisfaction  of 
extended  study  in  the  subject. 

It  is  at  once  evident  that  these  groups  are  not  sharply  marked  off 
from  each  other;  and  that  the  needs  of  the  first  group  are  shared 
by  the  others.  It  is,  moreover,  true  that  the  "  general  readers " 
represent  a  wide  range  of  interest.  The  committee  has  taken  all 
these  things  into  account,  and  still  believes  that  the  division  here 
made  will  prove  of  substantial  utility  in  arranging  the  offerings  of 
high-school  mathematics. 

3.  Comparative  values. — Out  of  the  conflict  of  topics  for  a  place  in 
the  program  there  emerges  one  general*  principle,  already  suggested 
in  these  pages,  which  is  being  increasingly  accepted  for  guidance  by 
students  of  education.  In  briefest  negative  terms :  No  item  shall  le 
retained  for  any  specific  group  of  pupils  unless,  in  relation  to 
other  items  and  to  time  involved,  its  (probable}  value  can  le  shou-n. 
So  stated  the  principle  seems  a  truism,  but  properly  applied  it 
proves  a  grim  pruning  hook  to  the  dead  limbs  of  tradition.  A 
final  method  of  ascertaining  such,  comparative  values  remains  to  bo 
worked  out;  but  the  feasibility  of  a  reasonable  application  of  the 
principle  will  hardly  be  denied.  In  accordance  with  this,  many 
topics  once  common  have  been  dropped  from  the  curriculum  and 
more  are  marked  to  go.  Thus  our  better  practice  has  ceased  to  in- 
clude the  Euclidian  method  of  finding  the  H.  C.  F..  because  the 
knowledge  of  this  method  is  nowhere  serviceable  in  life;  and  in 
secondary  algebra  itself  little  if  anything  else  depends  on  it.  In- 
deed, the  H.  C.  F.  itself  might  well  go,  as  it  is  used  almost  exclusively 
in  simplifying  fractions  made  for  the  purpose. 

In  a  full  discussion,  many  terms  of  the  statement  would  need  con- 
sideration. What  constitutes  value  is  probably  the  point  where  most 
questioning  would  arise.  The  committee  takes  this  term  in  its 
broadest  sense,  specifically  denying  restriction  to  a  "  bread  and  but- 


16  MATHEMATICS  IN  SECONDARY  EDUCATION. 

ter  "  basis  or  other  mere  material  utility,  though  affirming  that  re- 
munerative employment  is  normally  a  worthy  part  of  the  worthy 
life.  What  the  statement  then  in  fact  demands  is  (i)  that  the  value 
of  the  topic  be  not  a  mere  assumption — a  positive  case  must  be  made 
out;  and  (ii)  that  the  value  of  the  topic  so  shown  be  sufficiently  great 
in  relation  to  other  topics  and  to  the  element  of  cost  (as  regards 
time,  labor,  money  outlay,  etc.)  to  warrant  its  inclusion  in  the  cur- 
riculum. 

This  principle  of  exclusion  seems  especially  applicable  to  those 
items  which  now  remain  merely  as  a  heritage  from  the  past  and  to 
those  which  have  been  introduced  mainly  to  round  out  the  subject  or 
where  the  unity  of  the  subject  matter  has  been  found  in  the  con- 
tent itself  and  not  in  the  relation  of  the  content  to  the  needs  of  the 
pupil. 

In  offering  such  a  principle  for  guidance,  the  committee  considers 
that  it  is  merely  stating  explicitly  what  has  been  implicitly  assumed 
in  all  such  controversies.  The  committee  none  the  less  believes  that 
conscious  insistence  on  the  point  is  necessary  in  order  to  disclose 
whatever  indefensible  elements  may  be  in  our  present  program  of 
studies. 

4.  "Formed  discipline.''' — A  full  discussion  of  this  topic,  of  course, 
is  impossible  within  the  limits  of  this  paper.  Such  a  discussion  is, 
moreover,  for  our  purpose  unnecessary,  because  we  shall  wish  to 
use  only  the  most  general  conclusion,  in  which  there  is  substantial 
concurrence.  We  can  thus  avoid  the  niceties  of  elaboration,  about 
which  agreement  has  not  yet  been  reached.  The  older  doctrine  as- 
sumed uncritically  a  very  high  degrefe  of  what  we  now  call  "  general 
transfer  "  of  training.  Modern  investigation,  to  speak  generally,  re- 
stricts very  consi4erably  the  amount  of  transfer  which  may  reason- 
ably be  expected,  and  inquires  strictly  into  the  conditions  of  transfer. 
Under  the  older  doctrine  it  was  a  sufficient  justification  for  the  re- 
quiring of  any  subject  that  pupils  .should  gain  through  it  inerqased 
ability  in  the  use  of  any  important  "  faculty,"  because  the'increase 
in  ability  was  naively  assumed  to  mean  an  increase  in  the  equally 
naively  assumed  faculty  itself  and  would  accordingly  be  effective 
wherever  the  faculty  was  used.  As  pupils  shoAV  such  an  increase  of 
ability  in  one  or  more  "  faculties  "  by  the  simple  fact  of  learning  any 
new  subject,  the  convenience  of  this  older  doctrine  for  curriculum 
defense  is  evident.  When  this  old  psychological  doctrine  was  first 
called  in  question  by  scientific  measurement,  the  idea  gained  popular 
currency  that  all  transfer  was  denied.  No  such  claim  has  serious  sup- 
port. The  psychologists,  however,  have  so  far  found  it  difficult  to 
agree  upon  any  final  situation  as  to  the  amount  of  transfer  which  in 
any  particular  situation  may  be  a  priori  expected.  All  agree,  none 
the  less,  in  greatly  reducing  the  old  claim  both  as  to  the  amount  and 


ANALYSIS  OF  THE  SITUATION.  17 

as  to  the  generality  of  conditions  under  which  transfer  may  be  ex- 
pected. In  accordance  with  these  considerations  the  committee  has 
not  used  the  factor  of  "  formal  discipline  "  in  determining  the  con- 
tent of  the  mathematical  courses  to  be  recommended  in  this  report. 
5.  The  needs  of  the  several  groups. — With  these  several  princi- 
ples and  factors  before  us,  we  are  now  ready  to  consider  more  fully 
the  needs  of  the  several  groups  of  users  as  distinguished  above. 
[VVe  are  particularly  concerned  to  ask  whether  or  not  their  respective 
group  needs  are  compatible  with  one  introductory  course  to  be 
taken  in  common;  and  if  yes,  when  the  differentiation  from  such 
a  common  course  should  begin. 

(1)  The  "general  readers" — This  group  will  need    to    use    in 
"  practical "  fashion  but  little  of  mathematics  other  than  ordinary 
arithmetic.     As  general  readers,  however,  they  will  still  require  a 
certain  acquaintance  with  mathematical    language    and    concepts. 
Just  what  terms,  symbols,  and  concepts  would  meet  the  requirements 
of  this  group  will  have  to  be  determined  by  extensive  inductive 
studies.    Assuming,  however,  ordinary  arithmetic  and  mensuration, 
some  items  can  be  at  once  named  as  fairly  certain  to  be  included: 
How  to  interpret  and  evaluate  a  simple  literal  formula;  the  mean- 
ing and  use  of  an  algebraic  equation  of  one  unknown;  the  notion 
and  use  of  negative  numbers  in  such  simple  cases  as  temperature, 
latitude,  and  stock  fluctuations;  the  simpler  conception   of  space 
relations  (inductively  obtained)  ;  the  notion  of  function  (the  depend- 
ence of  one  quantity  upon  another) ;  the  graph  as  a  means  of  inter- 
preting statistical   information,   with  such   terms   as   average   and 
median. 

(2)  The  group  preparing  for  certain  trades. — Under  this  head 
the  committee  would  group  those  whose  use  of  "  practical "  mathe- 
matics is,  while  generally  quite  definite,  still  relatively  small — such, 
for  example,  as  machinists,  plumbers,  sheet-metal  workers,  .and  the 
like.     The  general  run  of  the  need  here  contemplated  can  be  gath- 
ered from  the  requirements  laid  down  for  machinists  in  one  of  the 
more  recent  vocational  surveys — simple  equations,  use  of  formulas, 
measurement  of  angles,  measurements  of  areas  and  volumes,  square 
root,  making  and  reading  of  graphs,  solution  of  right  triangles, 
geometry  of  the  circle.    Much  practice  would  of  course  be  necessary 
to  make  even  this  small  amount  of  mathematics  function  adequately. 

It  is  at  once  evident  that  if  no  more  algebra  is  needed  than  formu- 
las, simple  equations  and  the  graph,  and  no  more  geometry  than  is 
here  suggested,  then  the  ordinary  high-school  courses  in  these  sub- 
jects are  but  ill-adapted  to  the  needs  of  such  pupils.  It  would  seo.m 
to  follow  that  this  group  of  pupils  has  no  need  to  follow  courses  in 
mathematics  other  than  (i)  arithmetic,  (ii)  the  "interpretative" 


18  MATHEMATICS  IN  SECONDARY  EDUCATION. 

V  (introductory)  mathematics  discussed  above,  and  (iii)  the  special 
applications  of  these  to  the  specific  subject  matter  of  their  several 
specializations. 

This  group  might  then  well  study  in  common  with  the  preceding 
until  the  completion  of  the  work  there  laid  out.  The  presentation 
of  this  common  course  along  the  lines  previously  laid  down  (p.  11) 
would  well  harmonize  the  somewhat  "diverse  interests  of  the  two 
groups.  What  little  additional  content  and  whatever  practice  in 
specialized  application  this  second  group  might  need  could  then  be 
given  either  in  a  parallel  or  in  a  succeeding  course  (or  courses) 
especially  devised  for  that  purpose. 

(3)  The  group  preparing  for  engineering. — This  group  will  con- 
sist mostly  of  boys  intending  to  study  in  engineering  schools.  In 
contrast  with  the  two  preceding  groups,  appreciably  more  mathe- 
matics is  here  needed.  In  contrast  with  the  following  group,  there 
are  here  specific  aims  external  to  mathematics  itself  which  define  and 
limit  the  mathematical  knowledge  and  skill  needed.  Although 
recognizing  that  the  individual  teacher  will  require  a  certain  leeway 
as  regards  content  in  getting  his  class  effectively  to  wrork  at  any 
topic,  we  may  still  profitably  ask  as  to  the  minimum  content  fixed 
for  this  group  by  its  peculiar  needs. 

The  minimum  mathematical  content  suitable  for  the  use  of  this 
group  can  probably  best  be  secured  by  working  simultaneously  along 
two  lines:  First,  to  ascertain  inductively  what  mathematics  the 
engineer  needs  (including  experiment  to  find  out  what  part  of  this 
can  best  be  taught  in  the  secondary  school)  ;  second,  to  criticize  the 
existing  courses  to  see  what  they  lack  and  what  they  include  that  is 
useless  for  this  group.  It  is  much  to  be  hoped  that  necessary  in- 
ductive studies  and  experiments  along  the  first  line  of  procedure  may 
be  vigorously  pushed.  The  second  in  important  respects  waits  for 
the  first,  but  it  is  possible  from  certain  inherent  considerations  at 
once  to  exclude  some  matter  now  customarily  taught. 

Taking  the  customary  high-school  mathematics  as  a  basis  for  com- 
parison, we  find  at  least  three  principles  of  criteria  for  exclusion 
from  the  present  offerings:  (a)  Exclude  all  those  items  which  are 
not  themselves  to  be  directly  used  in  practical  situations  or  which  are 
not  reasonably  necessary  to  the  intelligent  mastery  or  use  of  such 
" practical"  items;  (£>)  exclude  all  involved  and  complicated  in- 
stances of  otherwise  useful  topics  or  applications  which  do  not  serve 
to  clarify  the  main  point  under  consideration;  (c)  exclude  all  such 
proofs  and  discussions  as  do  not  in  fact  help  the  pupil  to  an  intelli- 
gent use  of  the  topic.  It  is  probably  correct  to  say  that  these  exclu- 
sions relate  to  material  introduced  from  considerations  of  theory 
rather  than  of  intelligent  practical  mastery;  from  considerations  of 


ANALYSIS  OF  THE  SITUATION.  19 

the  pleasure  that  thcorizcrs  (teachers  mostly)  get  from  the  study  of 
mathematics  rather  than  from  a  conscious  purpose  to  give  that  famil- 
iarity and  grasp  which  the  future  practical  man  will  need. 

Under  the  head  of  (a)  topics  excluded  as  not  needed  in  this  group 
the  committee  would  mention  such  as  the  H.  C.  F.  and  the  L.  C.  M. ; 
operations  with  literal  coefficients  (except  for  a  few  formulas)  ;  radi- 
cal equations;  the  theory  of  exponents,  except  the  simplest  opera- 
tions with  fractional  and  negative  exponents  (these  to  be  retained  to 
give  meaning  to  logarithms  and  the  slide  rule);  operations  with 
imaginaries;  cube  root;  proportion  as  a  separate  topic  (the  simple 
equation  suffices) ;  the  progressions. 

Among  (b)  excluded  complex  applications  might  be  mentioned  the 
following:  All  lengthy  exercises  in  multiplication  and  division;  fac- 
toring beyond  the  simplest  instances  of  the  four  forms  (i)  ax-fay, 
(ii)  a2— b2,  (iii)  a2-j-2ab+b2,  (iv)  x2+(a+b)  x+ab;  all  but  the  sim- 
plest fractional  forms  (the  more  complicated  are  in  fact  given  to 
illustrate  factoring);  all  radicals  beyond  ^/ab  ^nd  VaT^rT;  simulta- 
neous equations  of  more  than  two  unknowns;  simultaneous  quad- 
ratics (except  possibly  a  quadratic  and  a  linear)  ;  the  clock,  hare  and 
hounds,  and  courier  problems  and  the  like ;  the  extended  formal  dem- 
onstrative geometry  of  our  ordinary  schools;  most  trigonometry  be- 
yond the  use  of  sine,  cosine,  and  tangent  in  triangle  work. 

(<?)  Proofs  excluded  or  deferred  are  mostly  cared  for  in  (a)  and 
(b).  The  chief  instances  in  the  past  (too  often  yet  remaining  for 
the  "  specializes  ")  have  been  the  distinction  between  negative  quan- 
tities and  negative  numbers,  the  (supposedly)  rigorous  generalizing 
of  amXan=m'"1j  the  proof  of  too  evident •  propositions  in  geometry, 
the  incommensurable  cases  in  geometry,  the  general  proof  of  sin 

(*+y). 

It  may  be  mentioned  in  this  connection  that  teachers  of  mathematics 
from  arithmetic  onward  only  too  frequently  deceive  themselves  as 
to  the  place  that  the  presentation  of  a  rigorously  logical  proof  plays 
in  bringing  conviction.  The  worth  of  a  sense  of  logical  cogency 
can  hardly  be  overestimated;  but  we  who  teach  not  infrequently 
overreach  ourselves  in  our  zeal  for  it.  The  teacher  of  introductory 
mathematics  can  well  take  lessons  from  the  laboratory,  whore  careful 
measurement  repeated  under  many  different  conditions  will  bring  a 
conviction  often  otherwise  unknown  to  the  pupil  who  is  not  gifted 
in  abstract  thinking.  Probably  in  most  instances  an  inductively 
reached  conviction  is  the  best  provocative  of  an  appetite  for  a  yet 
more  thoroughgoing  proof. 

Everything  so  far  points  to  one  common  introductory  course. 
With  this  group  as  with  the  preceding,  the  point  of  differentiation 
would  seem  to  come  at  the  end  of  the  interpretative  course  first  dis- 
cussed for  the  "  general  readers."  Whether  this  third  (engineering) 


20  MATHEMATICS   IN   SECONDARY  EDUCATION. 

group  should  proceed  further  in  common  with  the  fourth  group  (of 
specializes),  we  later  consider  further  in  common  with  the  fourth 
group. 

(4)  The  group  of  specializers. — This  group  will  include  those 
pupils,  both  boys  and  girls,  who  "  like  "  mathematics.  While  these 
best  of  all  could  continue  to  work  with  the  present  offerings,  the  con- 
siderations urged  under  the  discussion  on  presentation  suffice,  in  the 
committee's  opinion,  to  demand  even  for  this  group  a  far-reaching 
reorganization  of  practically  all  of  secondary  mathematics. 

.Since  we  are  here  planning  for  those  who  specialize  in  mathe- 
matics, we  are  not  called  upon — after  meeting  the  interpretative 
need — to  consider  any  external  demands  upon  mathematics,  but  only 
such  a  selection  and  arrangement  within  the  subject  itself  as  best 
furthers  the  mathematical  activity.  Hitherto  the  arrangement 
within  the  course  has  been  made,  as  we  saw  in  the  discussion  on  pres- 
entation, in  answer  to  considerations  rather  of  "  logical "  organiza- 
tion than  of  psychological  experiencing  and  growth.  The  results 
have  not  been  satisfactory.  Algebra,  geometry,  and,  to  a  lesser  de- 
gree, trigonometry  have  been  treated  as  separate  logical  entities, 
with  consequent  loss  to  the  pupil  of  both  interest  and  power.  The 
committee  thinks  that  the  selection  and  organization  should  be  made 
in  the  light  of  experiment  as  to  which  conceptions  do  in  fact  prove 
successive!}'  most  strategic  in  the  pupils'  continued  approach  to 
mathematical  power.  The  result  would  probably  take  a  form  some- 
what analogous  to  the  "  general  science  "  course  which  is  now  being 
worked  out  in  that  field. 

That  this  group  should  take  its  introductory  work  in  common  with 
the  others  has  perhaps  been  sufficiently  implied.  The  intelligent 
choice  of  a  specialty  could  hardly  precede  the  actual  experiencing  of 
taste  and  aptitude.  How  far  beyond  the  common  introductory  course 
this  group  should  go  in  company  with  the  third  (the  preliminary 
engineering)  is  not  easy  to  say.  In  all  but  the  largest  schools 
administrative  considerations  will  .probably  keep  the  two  together  in 
whatever  work  is  offered.  Where  numbers  and  funds  suffice,  differ- 
entiation may  well  begin  immediately  upon  the  completion  of  the 
common  introductory  course,  according  to  considerations  already 
laid  down.  In  that  case  the  preliminary  engineering  group  would 
get  their  mathematics  more  in  terms  of  engineering  content  and  sit- 
uations ;  those .  specializing  in  mathematics  would  get  theirs  more 
directly  in  terms  of  "  pure "  mathematics.  The  contents  of  such 
courses  could  well  differ  considerably. 

6.  Selecting  mathematical  ability. — From  the  point  of  view  both 
of  society  and  its  needs  and  of  the  individual  and  his  satisfactions,  it 
is  highly  desirable  that  ability,  or  the  lack  thereof,  be  disclosed  in 
order  that  intelligent  choice  may  be  made.  Mathematical  ability  as 


SUGGESTIONS  AS  TO  COURSES.  21 

expressed  in  mathematical  achievement  and  application  is  a  most 
powerful  agency  in  -advancing  civilization.  In  order  that  society 
may  profit  by  its  available  stock  of  mathematical  ability,  there  is 
urgent  need  of  some  process  that  shall  disclose  this  ability.  Anal- 
ogous considerations  demand  that  the  individual  learn  by  a  less 
costly  process  than  occupational  trial  what  degree  of  probable  success 
he  may  expect  from  an  occupation  in  which  mathematical  ability  is 
an  important  factor.  We  hope  much  from  further  psychological 
study  in  this  field  of  disclosing  specific  abilities,  but  as  matters  now 
stand  the  opportunity  in  the  high  school  for  trial  of  mathematical 
success  and  liking  is  at  least  one  important  part  in  the  disclosing 
of  mathematical  ability.  This  factor  must  be  taken  into  account  in 
arranging  the  introductory  work  in  mathematics. 

IV.  SUGGESTIONS  AS  TO  COURSES. 

Each  valid  consideration  in  the  foregoing  discussion  should  have 
its  effect  in  the  resulting  determination  of  the  mathematics  courses. 
Considerations  of  presentation  demanded  that  we  give  up  the  "  log- 
ical" arrangement  of  subject  matter,  especially  for  introductory 
work,  and  find  instead  an  organization  based  upon  the  successful  at- 
tack of  projects  and  problems  in  connection  with  which  the  pupils 
\  already  have  both  knowledge  and  potential  interest.  Four  groups  of 
1  pupils  judged  by  probable  destination  showed  four  types  of  mathe- 
matical needs:  (i)  The  "general  readers,"  whose  needs  lie  largely 
in  the  "interpretative"  function  of  mathematics;  (ii)  those  who, 
expecting  to  enter  trades,  would  have  a  small  but  still  definite  need 
for  "practical"  mathematics;  (iii)  those  who,  as  prospective  en- 
gineers, would  need  a  considerable  body  of  content  determined  by  the 
demands  of  engineering  study  and  practice;  (iv)  those  specializing  in 
mathematics  who  would  wish  a  content  determined  by  the  satisfac- 
tions inherent  in  the  activity  and  by  the  demands  of  further  study. 
From  considerations  of  comparative  values  nothing  should  enter  into 
the  curriculum  except  as  it  can  show  probable  value  in  relation  to 
other  topics  and  to  time  involved.  "  Formal  discipline "  was  not 
considered  by  the  committee  in  determining  the  content  of  courses 
to  be  recommended.  Care  should  be  given  that  at  an  early  stage 
mathematical  taste  and  ability  may  be  disclosed  so  as  to  allow  appro- 
priate choice  of  school  work  and  occupational  preparation.  It 
seemed  clear  that  a  new  introductory  course  should  be  offered  which 
all  the  students  should,  normally,  take  in  common.  College  entrance 
considerations,  except  as  inherently  cared  for  above,  are  deliberately 
disregarded. 

With  these  demands  before  us,  can  an  appropriate  school  procedure 
be  devised  and  feasibly  operated  ? 


22  MATHEMATICS  IN  SECONDARY  EDUCATION". 

The  task  certainly  is  great.  Nothing  short  of  extended  study  and 
experimentation  can  meet  the  situation.  The  committee  makes  the 
following  tentative  suggestions  as  possible  lines  along  which  research 
and  trial  might  prove  profitable. 

1.  The  ivork  of  the  junior  high  school. — It  seems  to  the  committee 
that  the  work  of  grades  7,  8,  and  9  should,  in  addition  to  whatever 
review  of  previous  arithmetic  may  be  necessary,  include — 

A.  A  body  of  processes  and  conceptions  commonly  called  arith- 
metic, where  the  study,  however,  is  of  social  activities — trade  or  other- 
wise— which  need  mathematics,  rather  than  of  mathematical  topics 
artificially  "motivated"  by  social  relationships.     As  a  constituent 
part  of  these  processes  the  committee  would  include  any  use  of  alge- 
bra or  intuitive  geometry  within  easy  reach  of  the  pupils  which  can 
prove  its  worth  by  actual  service  in  common  life  outside  of  the  school. 

B.  A  body  of  mathematical  symbols,  concepts,  information,  and 
processes — commonly  thought  of  as  belonging  to  algebra  and  geom- 
etry or  beyond— which  the  intelligent  general  reader  of  high  school 
or  college  standing  will  need  in  order  to  meet  the  demands  of  his 
social  and  intellectual  life.     As  a  part  of  this  content,  it  seems  safe  to 
suggest  the  ordinary  algebraic  symbols,  the  use  of  the  formula,  the 
simple  equation,  and  the  (statistical)  graph. 

C.  The  opportunity  for  at  least  a  preliminary  testing  of  mathe- 
matical taste  and  aptitude.1 

D.  Such  additional  content — relatively  small  in  amount — as  may 
be  needed  to  make  effective  the  teaching  of  the  foregoing. 

The  appropriate  contents  of  parts  A  and  B  can  be  fixed  only  by  a 
carefully  made  inductive  study  of  the  demands  as  they  actually  ex- 
ist; the  contents  of  parts  C  and  D,  only  after  extended  experimenta- 
tion. The  contents  of  B,  C,  and  D,  respectively,  the  committee 
judges  to  be  in  the  descending  order  of  size  and  importance.  Pend- 
ing the  scientific  determination  of  these  several  contents,  the  com- 
mittee feels  that  a  wide  diversity  of  offerings  is  to  be  welcomed  as 
a  sign  of  healthy  variation  likely  to  promote  progress. 

Just  what  course  groupings  of  the  foregoing  should  be  made  must 
likewise  be  for  some  time  a  matter  of  experimentation.  Some  will 
wish  to  consider  the  whole  three  years'  work  as  one  unity,  the 
various  items  being  presented  in  such  connections  among  them- 
selves and  with  the  situations  of  application  as  good  teaching  may 
suggest.  Others  will  wish  to  give  A  in  grade  7,  and  devote  grades 
8  and  9  to  an  extended  treatment  of  B,  C,  D.  Still  others  will  give 
two  years  to  A,  probably  reducing  the  weekly  time  allowance,  and 
in  the  ninth  grade  concentrate  on  B,  C,  D. 

1  We  have  grounds  for  hoping  that  psychological  tests  may  prove  of  material  assistance 
in  this  connection. 


SUGGESTIONS  AS  TO   COURSES.  23 

Especial  attention  is  called  to  the  course  to  be  made  up  of  B,  C,  and 
D  (whether  extending  through  one  year  or  two).  This  is  the  com- 
mon introductory  course  referred  to  many  times  in  these  pages.  It 
is  assumed  that  as  a  rule  all  pupils  would  take  it  (or  at  least  begin  on 
it) ,  and  that  no  further  mathematics  would  be  customarily  required 
for  college  entrance^1 

2.  Trade   mathematics. — For   the   groups   of   small   but   definite 
"  practical "  use  the  committee  judges  that  the  foregoing  will  com- 
monly suffice  so  far  as  concerns  specific  mathematical  content.     In 
some  of  the  trade  curriculums,  however,  it  will  be  necessary  to  pro- 
vide a  specific  course  (or  courses)  in  which  sufficient  practice  in  the 
trade  application  can  be  found.    The  more  directly  such  courses  can 
be  connected  with  the  work  of  application  the  better. 

3.  Preliminary    engineering. — For    the    preliminary    engineering 
group  there  are,  after  the  common  introductory  course,  several  possi- 
bilities.   One  would  be  to  have  this  group  work  as  heretofore  with  the 
" specializes "   (see  4  below).     This  is  perhaps  less  desirable,  but 
will  probably  continue  for  some  time  to  come  as  the  more  usual  pro- 
cedure, especially  in  the  secondary  school  of  not  more  than  moderate 
size  and  income.    Another  possibility  would  be  to  construct  a  course 
specifically  for  this  group  from  a  careful  study  of  the  specific  de- 
mands of  their  future  work  (see  pages  18  and  19  above).     Such'  a 
content  could  then  be  given  according  to  the  principles  of  presenta- 
tion discussed  earlier  (see  p.  11).    Here  again  experimentation  will 
be  necessary  to  develop  an  effective  organization  and  procedure. 
Such  experimentation  may  be  expected  to  show  a  wide  range  of  va- 
riation— at  the  one  extreme  an  approximation  to  the  old  formal 
"logical";  at  the  other,  an  effort  to  make  all  mathematics  teaching 
purely  incidental  to  other  work.    And  again,  a  wide  diversity  is  at 
the  first  a  healthy  indication. 

4.  For  the   specializes. — Where  numbers   and   income  warrant, 
there  should  be  elective  work  during  the  grades  10,  11,  and  12  for 
those  specializing  in  mathematics.     There  is  need,  as  previously 
stated,  to  reorganize  the  customary  offerings  for  these  years  in  such 
a  way  as  to  displace  a  presentation  based  on  classification  for  a  pre- 
sentation based  on  experimentally  determined  conditions  in  growth, 
in  interest,  and  power. 

Such  a  reorganization  will  naturally  run  across  the  lines  of  divi- 
sion heretofore  maintained,  and  will  probably  anticipate  certain 
conceptions  and  procedures  confined  now  to  analytics  and  calculus. 
[Where  the  preliminary  engineering  group  is  included  with  this 

*Any  pupil  would  of  course  be  permitted  to  offer  for  entrance  credit  any  other  mathe- 
matics he  had  elected  in  his  secondary  school.  It  is,  moreover,  probable  that  certain 
college  courses  open  to  freshmen  would  specify  as  a  necessary  prerequisite  an  amount  of 
mathematics  greater  than  that  here  included  in  this  "  general  readers'  "  course. 


24  MATHEMATICS  IN  SECONDARY  EDUCATION. 

group  it  may  prove  advisable  to  utilize  to  a  considerable  extent 
problems  and  projects  from  the  natural  sciences  to  give  a  certain 
desirable  concreteness  of  thinking.  This  may  well  result  in  benefit 
to  all  concerned.  With  progress  in  the  work  should  come,  however, 
for  the  group  of  specializers  an  increased  interest  in  and  desire  for 
mathematics  on  its  own  account.  Such  a  reorganization  as  is  sug- 
gested above  would  probably  reduce  in  appreciable  degree  the  quan- 
tity of  formal  demonstrative  geometry,  a  result  that  the  committee 
anticipates  with  equanimity.  It  seems  probable  that  by  suitable 
experimentation  a  new  course  can  be  worked  out  which  will  prove 
more  alluring  to  the  pupil  and  at  the  same  time  furnish  a  better 
introduction  to  the  further  study  of  the  subject.  Various  efforts 
tending  to  corroborate  this  belief  have  already  been  made  both  in 
this  country  and  abroad. 

It  may  be  asked  whether  all  secondary  schools  should  try  to  make 
full  offerings  of  the  courses  here  suggested.  The  committee  thinks 
not.  It  will  expect  that  work  substantially  equivalent  to  that  here 
suggested  for  the  grades  7,  8,  and  9  will  everywhere  be  offered;  that 
the  trade  courses  will  naturally  be  restricted  to  trade  curriculums ;  but 
that  the  elective  work  for  the  senior  high  school  may  be  restricted 
in  small  schools  where  the  income  is  not  large.  It  seems  probable 
that  the  relative  reduction  attending  elective  mathematics  in  the  col- 
lege will  extend  itself  similarly  to  the  secondary  school.  The  com- 
mittee in  conclusion  deprecates  the  continued  disposition  on  the  part 
of  some  colleges  unwisely  to  dictate  the  contents  of  courses  in 
secondary  schools.  It  feels  that  such  a  usurpation  of  power  operates 
to  prevent  the  secondary  school  from  making  the  most  intelligent 
adaptation  of  its  work  to  the  needs  of  its  pupils. 


o 


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MAR  19 1942 


9  Ma/611 


D 


LD  21-10 


